Recently,it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the sim-ple density wave propagation example ...Recently,it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the sim-ple density wave propagation example for the compressible Euler equations.The issue is related to missing local linear stability,i.e.,the stability of the discretization towards per-turbations added to a stable base flow.This is strongly related to an anti-diffusion mech-anism,that is inherent in entropy-conserving two-point fluxes,which are a key ingredi-ent for the high-order discontinuous Galerkin extension.In this paper,we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations.Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation.We present the full theoretical derivation,analysis,and show corresponding numerical results to underline our findings.In addition,we characterize numerical fluxes for the Euler equations that are entropy-conservative,kinetic-energy-preserving,pressure-equilibrium-preserving,and have a density flux that does not depend on the pressure.The source code to reproduce all numerical experiments presented in this article is available online(https://doi.org/10.5281/zenodo.4054366).展开更多
In this paper,we develop an entropy-conservative discontinuous Galerkin(DG)method for the shallow water(SW)equation with random inputs.One of the most popular methods for uncertainty quantifcation is the generalized P...In this paper,we develop an entropy-conservative discontinuous Galerkin(DG)method for the shallow water(SW)equation with random inputs.One of the most popular methods for uncertainty quantifcation is the generalized Polynomial Chaos(gPC)approach which we consider in the following manuscript.We apply the stochastic Galerkin(SG)method to the stochastic SW equations.Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore.The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations.By transforming the system using Roe variables,the hyperbolicity can be ensured and an entropy-entropy fux pair is known from a recent investigation by Gerster and Herty(Commun.Comput.Phys.27(3):639–671,2020).We use this pair and determine a corresponding entropy fux potential.Then,we construct entropy conservative numerical twopoint fuxes for this augmented system.By applying these new numerical fuxes in a nodal DG spectral element method(DGSEM)with fux diferencing ansatz,we obtain a provable entropy conservative(dissipative)scheme.In numerical experiments,we validate our theoretical fndings.展开更多
The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations.The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in...The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations.The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts(SBP)difference boundary closure of(Gerritsen and Olsson in J Comput Phys 129:245-262,1996;Olsson and Oliger in RIACS Tech Rep 94.01,1994;Yee et al.in J Comp Phys 162:33-81,2000).Sj?green and Yee(J Sci Comput)recently proved that the entropy split method is entropy conservative and stable.Stand-ard high-order spatial central differencing as well as high order central spatial dispersion relation preserving(DRP)spatial differencing is part of the entropy stable split methodol-ogy framework.The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives.Due to the construction,this conservative numerical flux requires higher oper-ations count and is less stable than the original semi-conservative split method.However,the Tadmor entropy conservative(EC)method(Tadmor in Acta Numerica 12:451-512,2003)of the same order requires more operations count than the new construction.Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative,a modified nonlinear filter approach of(Yee et al.in J Comput Phys 150:199-238,1999,J Comp Phys 162:3381,2000;Yee and Sj?green in J Comput Phys 225:910934,2007,High Order Filter Methods for Wide Range of Compressible flow Speeds.Proceedings of the ICOSAHOM09,June 22-26,Trondheim,Norway,2009)is proposed in conjunction with the entropy split method as the base method for problems containing shock waves.Long-time integration of 2D and 3D test cases is included to show the com-parison of these new approaches.展开更多
The maximum entropy principle(MEP) is one of the first methods which have been used to predict droplet size and velocity distributions of liquid sprays. This method needs a mean droplets diameter as an input to predic...The maximum entropy principle(MEP) is one of the first methods which have been used to predict droplet size and velocity distributions of liquid sprays. This method needs a mean droplets diameter as an input to predict the droplet size distribution. This paper presents a new sub-model based on the deterministic aspects of liquid atomization process independent of the experimental data to provide the mean droplets diameter for using in the maximum entropy formulation(MEF). For this purpose, a theoretical model based on the approach of energy conservation law entitled energy-based model(EBM) is presented. Based on this approach, atomization occurs due to the kinetic energy loss. Prediction of the combined model(MEF/EBM) is in good agreement with the available experimental data. The energy-based model can be used as a fast and reliable enough model to obtain a good estimation of the mean droplets diameter of a spray and the combined model(MEF/EBM) can be used to well predict the droplet size distribution at the primary breakup.展开更多
For Hawking radiation, treated as a tunneling process, the no-hair theorem of black hole together with the law of energy conservation is utilized to postulate that the tunneling rate only depends on the external quali...For Hawking radiation, treated as a tunneling process, the no-hair theorem of black hole together with the law of energy conservation is utilized to postulate that the tunneling rate only depends on the external qualities(e.g., the mass for the Schwarzschild black hole) and the energy of the radiated particle. This postulate is justified by the WKB approximation for calculating the tunneling probability. Based on this postulate, a general formula for the tunneling probability is derived without referring to the concrete form of black hole metric. This formula implies an intrinsic correlation between the successive processes of the black hole radiation of two or more particles. It also suggests a kind of entropy conservation and thus resolves the puzzle of black hole information loss in some sense.展开更多
Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme.The essential feature is that the momentum flux should be of the fo...Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme.The essential feature is that the momentum flux should be of the form ■ are any consistent approximations to the pressure and the mass flux.This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging.Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes.As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed.These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse.We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition.Secondly,we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes.These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme.For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows.Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.展开更多
This paper develops the high-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamic equations.The schemes are built on the entropy conservative flux and t...This paper develops the high-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamic equations.The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory(WENO)technique as well as explicit Runge-Kutta time discretization.The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair.As soon as the entropy conservative flux is derived,the dissipation term can be added to give the semidiscrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function.The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes.Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.展开更多
The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entro...The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entropy condition.Until 2020,for one-dimensional gas dynamics equations,the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate(AA)flux and the entropy conservative(EC)flux developed by Maire et al.was used.This hybridized CL scheme satisfies the entropy condition;however,it is under-entropic in the part zones of rarefaction waves.Moreover,the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves.Another disadvantage of this scheme is that it is of only first-order accuracy.In this paper,we firstly construct a modified entropy conservative(MEC)flux which can damp effectively numerical oscillations in simulating strong rarefaction waves.Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows.This new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic,we propose using the third-order TVD-type Runge-Kutta time discretization method.Based on the new hybridized flux,we develop the second-order CL scheme that satisfies the entropy condition.Finally,the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.展开更多
基金Funded by the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)under Germany’s Excellence Strategy EXC 2044-390685587Mathematics Münster:Dynamics-Geometry-Structure.Gregor Gassner is supported by the European Research Council(ERC)under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme,ERC Grant Agreement No.714487.
文摘Recently,it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the sim-ple density wave propagation example for the compressible Euler equations.The issue is related to missing local linear stability,i.e.,the stability of the discretization towards per-turbations added to a stable base flow.This is strongly related to an anti-diffusion mech-anism,that is inherent in entropy-conserving two-point fluxes,which are a key ingredi-ent for the high-order discontinuous Galerkin extension.In this paper,we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations.Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation.We present the full theoretical derivation,analysis,and show corresponding numerical results to underline our findings.In addition,we characterize numerical fluxes for the Euler equations that are entropy-conservative,kinetic-energy-preserving,pressure-equilibrium-preserving,and have a density flux that does not depend on the pressure.The source code to reproduce all numerical experiments presented in this article is available online(https://doi.org/10.5281/zenodo.4054366).
文摘In this paper,we develop an entropy-conservative discontinuous Galerkin(DG)method for the shallow water(SW)equation with random inputs.One of the most popular methods for uncertainty quantifcation is the generalized Polynomial Chaos(gPC)approach which we consider in the following manuscript.We apply the stochastic Galerkin(SG)method to the stochastic SW equations.Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore.The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations.By transforming the system using Roe variables,the hyperbolicity can be ensured and an entropy-entropy fux pair is known from a recent investigation by Gerster and Herty(Commun.Comput.Phys.27(3):639–671,2020).We use this pair and determine a corresponding entropy fux potential.Then,we construct entropy conservative numerical twopoint fuxes for this augmented system.By applying these new numerical fuxes in a nodal DG spectral element method(DGSEM)with fux diferencing ansatz,we obtain a provable entropy conservative(dissipative)scheme.In numerical experiments,we validate our theoretical fndings.
基金support from the NASA TTT/RCA program for the second author is grate-fully acknowledged.
文摘The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations.The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts(SBP)difference boundary closure of(Gerritsen and Olsson in J Comput Phys 129:245-262,1996;Olsson and Oliger in RIACS Tech Rep 94.01,1994;Yee et al.in J Comp Phys 162:33-81,2000).Sj?green and Yee(J Sci Comput)recently proved that the entropy split method is entropy conservative and stable.Stand-ard high-order spatial central differencing as well as high order central spatial dispersion relation preserving(DRP)spatial differencing is part of the entropy stable split methodol-ogy framework.The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives.Due to the construction,this conservative numerical flux requires higher oper-ations count and is less stable than the original semi-conservative split method.However,the Tadmor entropy conservative(EC)method(Tadmor in Acta Numerica 12:451-512,2003)of the same order requires more operations count than the new construction.Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative,a modified nonlinear filter approach of(Yee et al.in J Comput Phys 150:199-238,1999,J Comp Phys 162:3381,2000;Yee and Sj?green in J Comput Phys 225:910934,2007,High Order Filter Methods for Wide Range of Compressible flow Speeds.Proceedings of the ICOSAHOM09,June 22-26,Trondheim,Norway,2009)is proposed in conjunction with the entropy split method as the base method for problems containing shock waves.Long-time integration of 2D and 3D test cases is included to show the com-parison of these new approaches.
文摘The maximum entropy principle(MEP) is one of the first methods which have been used to predict droplet size and velocity distributions of liquid sprays. This method needs a mean droplets diameter as an input to predict the droplet size distribution. This paper presents a new sub-model based on the deterministic aspects of liquid atomization process independent of the experimental data to provide the mean droplets diameter for using in the maximum entropy formulation(MEF). For this purpose, a theoretical model based on the approach of energy conservation law entitled energy-based model(EBM) is presented. Based on this approach, atomization occurs due to the kinetic energy loss. Prediction of the combined model(MEF/EBM) is in good agreement with the available experimental data. The energy-based model can be used as a fast and reliable enough model to obtain a good estimation of the mean droplets diameter of a spray and the combined model(MEF/EBM) can be used to well predict the droplet size distribution at the primary breakup.
基金Supported by National Natural Science Foundation of China and the National Fundamental Research Programs of China under Grant Nos. 10874091 and 2006CB921205
文摘For Hawking radiation, treated as a tunneling process, the no-hair theorem of black hole together with the law of energy conservation is utilized to postulate that the tunneling rate only depends on the external qualities(e.g., the mass for the Schwarzschild black hole) and the energy of the radiated particle. This postulate is justified by the WKB approximation for calculating the tunneling probability. Based on this postulate, a general formula for the tunneling probability is derived without referring to the concrete form of black hole metric. This formula implies an intrinsic correlation between the successive processes of the black hole radiation of two or more particles. It also suggests a kind of entropy conservation and thus resolves the puzzle of black hole information loss in some sense.
文摘Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme.The essential feature is that the momentum flux should be of the form ■ are any consistent approximations to the pressure and the mass flux.This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging.Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes.As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed.These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse.We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition.Secondly,we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes.These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme.For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows.Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.
基金supported by the Special Project on High-performance Computing under the National Key R&D Program(No.2016YFB0200603)Science Challenge Project(No.TZ2016002)the National Natural Science Foundation of China(Nos.91630310 and 11421101),and High-Performance Computing Platform of Peking University.
文摘This paper develops the high-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamic equations.The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory(WENO)technique as well as explicit Runge-Kutta time discretization.The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair.As soon as the entropy conservative flux is derived,the dissipation term can be added to give the semidiscrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function.The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes.Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.
基金Nation Key R&D Program of China(Grant No.2022YFA1004500)and NSFC(Grant No.12072043).
文摘The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entropy condition.Until 2020,for one-dimensional gas dynamics equations,the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate(AA)flux and the entropy conservative(EC)flux developed by Maire et al.was used.This hybridized CL scheme satisfies the entropy condition;however,it is under-entropic in the part zones of rarefaction waves.Moreover,the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves.Another disadvantage of this scheme is that it is of only first-order accuracy.In this paper,we firstly construct a modified entropy conservative(MEC)flux which can damp effectively numerical oscillations in simulating strong rarefaction waves.Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows.This new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic,we propose using the third-order TVD-type Runge-Kutta time discretization method.Based on the new hybridized flux,we develop the second-order CL scheme that satisfies the entropy condition.Finally,the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.
